Ú Bruguieres, A. Virelizier, A. [4] Á «Î µà Monoidal

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1 40 2 Æ Vol.40, No ADVANCES IN MATHEMATICS April, ) T- ÆÖ Ñ Ó 1,, 2 (1. È Ä È ; 2. È Ä È Ì. ½ A- (coring) T- (comonad) ( ± A º ¼ T º (monad)).» ³¹ (firm) µ ³ Frobenius ² ¾ ³ ¾ Ò T- Frobenius MR(2000) Ð 18D10; 16W30 / «Đ O154.1 ßÚ A (2011) Moerdijk, I. [1] Á Î Ã Í Hopf ¼Ç Hopf ¼Ç Ã Í ¼Ç µ Å¼Ç ²Ï Á ³ Hopf ¼Ç ƹ Hopf ¹ À 2006 El Kaoutit, L. Vercruysse, J. [2] Á Î Ç Æ Grothendieck à Á º¼Ç º¾Ã Ù ÍÚ ³ Æ ß Ç º¼Ç ØÙ Grothendieck Ã Í º¼Ç º¼ÇÍ º¾Ã Á ³ Ã ß «ÐÇ ½º¼Çµ º Ç ºÖÆ Î Ï± É Î±³ 2006 Gómez-Torrecillas, J. [3] Á Î º¼Ç ÜÍ Ç«ÐÀ κ¼Ç Æ F : C D, G : D C «ÐÇ Á C, D Îà C = GF à C Á º¼Ç Î º¼ÇÓÏ Í Í À Æ º¾Ã ³ Ã ß «ÐÇ Æ Ô º¼ÇÓÏ ÅÆ ³¹ Í Í Galois º¾  À Ú Bruguieres, A. Virelizier, A. [4] Á «Î µà Monoidal à Á Hopf ¼Ç ¼ Æ¼Ç ½ Ý Ð Hopf ¹ Æ Û Hopf ¼ÇÍ ¼Ç º¼Ç ¹ º¹ Û Beck, J. [5] Á ¹ A º¹ C λ : C k A A k C, ϱ ¼Ç T º¼Ç G Ò Æ ψ : TG GT, ³Æ ³¹ ± ¾Ã Mesablishvili, B. [6] Á Monoidal à Á ³Í Æ Å ¾Á Æ Û ¼ ÆÃ Í [7] ¼Ç º¼Ç À ¾Ã Í Î Ç ¼ ÎÇ «ÐÇ [8] Á Caenepeel, S. Á T- º¹ º ÆÇ ÆÇ ¹ ½º¹ À Òº ß Ý½Õ [9] A- Í µ À A- Í ÌÄ ÊÌ Ä ÕÅ ÜË ÕÅÆÄ (No ); ¾ Ñ Â ÕÅÆÄ È Ë µâ ÆÅÅ (No. RCZX200735) dairx129@163.com.

2 2 ºÇ Ý T-»½È Ç ÞØ 157 Æ Ý½Õ [10] [4] Hopf ¼Ç À Ô²Íϱ º Ö ºÁ º º Ê ¼Ç ¼Ç T T- ¾Ã Ç ß ¼Ç T º T- ¾Ã ßĐ Ï Í Ô²Í Ï± (firm) ¼Ç Frobenius Ç Ý½Õ ±½º¼Ç Æ Ý Î½Õ 1 T- Ý Þ Ã ¼Ç T- º¼Ç (T,C)- ¾Ã ΫÐÇ Ç«ÐÀ Ö ¼«[7]. 1.1 [4] C à ÎÃ Đº ÊÚÓÇEnd(C) Î Í Monoidal à C Í ¼Ç Ê (T,µ,η), ÁT : C C ÎÇ µ : T 2 T η : id C T Ê Ð µ X T(µ X ) = µ X µ T(X) ; µ X η T(X) = id T(X) = µ X T(η X ), X obj(c). (T,µ,η) C Í ¼Ç T- ¾ (M,r), Á M obj(c) r : T(M) M C Á ÓÏ r T(r) = r µ M ; r η M = id M. (M,r), (N,s) Î T- ¾ ÓÏ f Hom(M,N) T- É f r = s T(f). f T- ¾ÓÏ 1.2 [3] (G,,ε) à C Í º¼Ç ÉÇ G : C C Ê Ð : G G 2 ε : G id C, ¹Í G = G ; εg = Gε = id G. (G,,ε) à C Í º¼Ç G- º¾  (M,m), M obj(c) m : M GM à C Á ÓÏ M m = Gm m; ε M m = id M. (M,m), (N,n) G- º¾ ÓÏ g Hom(M,N) g- º É n g = Fg m. ³ Ð g G- º¾ÓÏ Ù 1.3 C à ΠMonoidal à A C Á À A? C ÊÚÓÇ (A?)(X) = A X (A?)(f) = id A f, Á X C Á À f C Á ÓÏ m : A A A u : 1 A C Á ÓÏ µ X = m id X η X = u id X. (A?,µ,η) C Í ¼Ç «(A,m,u) C Á ¹ [4]. Ú C? C Í º¼Ç «C à C Í º¹ 1.4 [3] A A- ¾ M É Ö²Ï M : M AA M Ï ²ÏÙ d + M : M M A A. Ï A- ¾ Ö²Ï ²ÏÆ Ù M d M. É A = A, d+ A = d A, ³ A 1.5 [3] A A- Í C ѵ A- ¾ Monoidal à Á º¹ Ø A- ¾ C, A- ¾ÚÓ C : C C A C, ε C : C A ¹Í (C C ) C = ( C C) C ; (ε C A C) C = d C ; (C ε C) C = d + C. 1.3, Đºµ 1.6 (T,µ,η) à C Í ¼Ç T- ¾ M obj(c) T- ¾ É M : MT M Ï Ù d+ M : M MT. Ú Ï T- ¾ M obj(c) Ï T- ¾ É M d M : M TM. : TM M Ï Ù À¹Ã C Í ¼Ç (T,µ,η) Monoidal à End(C) Á T- ¾ Á r = µ : TT T. É T = T, d+ T = d T, ¼Ç T ¼Ç 1.7 (T,µ,η) à C Í ¼Ç C End(C), T- º¼Ç C C T-

3 ¾ÓÏ T- ÓÏ C : C C 2, ε C : C T ¹Í C C C = C C C ; Cε C C = d + C ; ε C C C = d C. ÛÍ T- º¼Ç C T- ¾Ã Í º¼Ç Ù 1.8 T Monoidal à C Í ¼Ç T- º¼Ç G µ g(øê Ð g, ¹ Í X obj(c), (X) g(x) = g 2 (X);ε(X) g(x) = X, «T ÏÓ G- º¾ ÅÛ Å T G- º¾ ³ ρ : T TG, g µ ρg ρ = TGg Tg = T(Gg g) = T( )g = Tgg = T g; Tε ρ = Tε Tg = T(ε g) = T. Æ 1.9 C T- º¼Ç T ¼Ç (C,T)- º¾Ã C M T À (M,m), Á m : M CM Ê ÓÏ ¹Í C M m = Cm m; ε C M m = d M. ÓÏ T- ÓÏ f : (M,m) (N,n), f : M N ¹Í n f = Cf m. (T, C)- ¾Ã Ü 1.10 ß Ç F : C M T M T µ «ÐÇ G : M T C M T, Á G(N) = CN, G(f) = Cf (G,δ,ǫ) T- ¾Ã Í ² κ¼Ç (C,G)- º¾Ã C M G À (M,m,n), Á M M T, m : M CM, n : M MG, C M m = Cm m; ε C M m = d M ; Mδ n = ng n; Mǫ n = id M. µ Å Ð mg n = Cn m. ÓÏ f : (M,m,n) (M,m,n ), Á (M,m,n) C M G, (M,m,n ) C M G, f : M M Ê ¹Í m f = Cf m; n f = fg n. Ü 1.12 ß Ç Φ : C M G C M T µ «ÐÇ Ψ : C M T C M G, Á Ψ(M,n ) = (M G,m G,M ǫ); Ψ(f) = fg. (F,G ) (Φ,Ψ) Ç«ÐÀ [7]. 2 Frobenius ÐÕ 2.1 [8] C, C, D à ÊÎà F : C D Ç F Ê Ð f : Hom C (, ) Hom D (F( ),F( )), Ç F Æ É f Ʊ Ø Ê Ð p : Hom D (F( ),F( )) Hom C (, ) p f = 1 HomC (, ). 2.2 [8] F : C D Ç É Ç G : D C Ú F Ï«ÐÇ F «ÐÇ F Frobenius Ç (F,G) C D Frobenius ÇÀ ¹ [8] Á ½Õ Đºµ Ü 2.3 (T,µ,η) à C Í ¼Ç C T- º¼Ç ½Õ Ý (a) ß Ç F : C M T M T Frobenius Ç (b) ß Ç G : T M C T M Frobenius Ç (c) (T,T)- ÓÏ e : T C (C,C)- ¾ÓÏ π : CC C π Cη = C, π ηc = C. ÅÛ (c) (a) ½Õ 1.6, F : C M T T M µ «ÐÇ G : M T C M T, G(N) = CN, G(f) = Cf. G : M T C M T F Ï«ÐÇ

4 2 ºÇ Ý T-»½È Ç ÞØ 159 ϕ : I FG, ψ : GF I, φ ψ Æ «Ð ¼Ð º¼Ð φ ψ Ê Ð Fψ ϕf = I, ψg Gϕ = I( Ö ¼ ). (a) (c) F Frobenius Ç ϕ : I FG, ψ : GF I Æ «Ð ¼Ð º¼ Ð π = ψc : CC C, e = C ϕt : T C, π, e ¹Í (c) Á Ð Ú (b) (c). Ø 2.4 (T,µ,η) à C Í ¼Ç C T-º¼Ç ß Ç F : T M C M T C Cγ CC = C γc C C. Æ «Ê Ðγ : C 2 T γ C = ε C ; ÅÛ F : T M C M T, µ «ÐÇ G : M T T M C, ³¼Ð φ : id TM C GF, à N T M C, φ N : N GFN. ¹ F Æ (T,T)- Ê Ð ν : GF id TM C ν N φ N = id N. ¹º¼ÇC C- º¾ φ C = C, Áµ ± ν C : C 2 C ν C C = id C. γ = ε C ν C : C 2 T. γ (T,T)- γ Ê Ð γ C = ε C ; C Cγ CC = µ Tγ = µ Tε C Tν C = ε C C Tν C = ε C ν C C C = ε C ν C C C = γ C C; µ γt = µ ε C T ν C T = ε C C ν CT = ε C ν C C C = ε C ν C C C = γ C C Ø. γ (T,T)- Àà (T,T)- ²Ï f : M N, µ Tf γ M = TF ε C M ν C M = ε C N Cf ν C M = ε C N γ C N C 2 f = γ N C 2 f; Ø γ Ê Ð γ C = ε C ν C C = ε C. Î C γc C C = C ε CC ν C C C C = C ε CC C ν C = C d C ν C = id C ν C = C d+ C ν C = C Cε C ν C = C Cε C Cν C C C = É γ : C 2 T ¹ÍÕ ν M : MC M ν M = M Mγ ρ MC. ν M (T,T)- ν M º ν M Ê Ð ν M φ M = id M. M Tν M = T M M TMγ Tρ MC = M M T TMγ Tρ MC = M Mγ M C2 Tρ M C = M Mγ ρ MC M C = ν M M C; M ν MT = M M T MγT ρ MCT = M Mγ MC C ρ MCT = M Mγ ρ MC M C = ν M M C. Ø ν M (T,T)- ρ M ν M = ρ M = M C MγC MC C ρ M C = M Ø ν M º M Mγ ρ MC = M C ρ MT Mγ ρ M C = M C MCγ M C ρ M C M C MγC ρ MC 2 ρ M C = ν M ρ M C. Àà (T,T)- ÓÏ f : M N, µ f ν M = f M Mγ ρ MC = N ft Mγ ρ MC = N Nγ fc2 ρ M C = N Nγ ρ NC fc = ν N fc. Ø ν M Ê Ð Î ν M φ M = M Mγ ρ MC φ M = M Mγ M C ρ M = M Mε C ρ M = M d+ M = id M. Ø 2.5 (T,µ,η) à C Í ¼Ç T- º¼Ç C º Æ «ß Ç F : T M C M T Æ ÅÛ É C º Æ (T,T) ÓÏ π : C 2 C, Cπ C C = C π = πc C C ; π C = C. γ : C 2 T γ = ε C π. γ C = ε C ; C Cγ CC =

5 γ C = ε C π C = ε C C = ε C. C Cγ CC = C Cε C Cπ C C = C Cε C C π = = C d C π = C ε CC C π = C ε CC πc C C = C d+ C π = id C π 2.4, ß Ç F : T M C M T Æ π : C 2 C π = C Cγ CC, π (T,T)- π C- º π C Ʊ Ø C º Æ ÌÍ µ» Ø 2.6 (T,µ,η) à C Í ¼Ç C T- º¼Ç ½Õ Ý (a) C º Æ i.e. (T,T)- Ê Ð π : C 2 C, π = πc C = Cπ C; π = id C. (b) (T,T)- Ê Ð γ : C 2 T C Cγ CC = C γc C C, γ C = ε C. (c) ß Ç F : C M T M T Æ (d) ß Ç G : T M C T M Æ (e) ß Ç G : C T MC T TM T Æ Ô [1] Moerdijk, I., Monads on tensor categories, J. Pure Appl. Algebra, 2002, 168: [2] El Kaoutit, L., Vercruysse, J., Cohomology for Bicomodules, Separable and Maschke Functors, ArXiv: math.ra/ [3] Gómez-Torrecillas, J., Comonad and Galois corings, Applied Categorical Structure, 2006, 14(5-6): [4] Bruguieres, A., Virelizier, A., Hopf Monad, Adv. Math., 2007, 215(2): [5] Beck, J., Distributive laws, Lect. Notes Math., 1969, 80: [6] Mesablishvili, B., Entwining structures in monoidal categories, Journal of Algebra, 2008, 319(6): [7] Þ»ÈÅ ¾É ¼¾É À 2008, 51(5): [8] Caenepeel, S., Militaru, G., Zhu S. L., Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations, Vol.1787, Springer, [9] Brzeziński, T., Wisbauer, R., Corings and Comodules, Cambrige: Cambrige University Press, [10]»ÈÅ µ Þ ¼ Ë ¾É 2010, 53(5): The Equivalent Conditions of Separable T-comonads DAI Ruixiang 1, WANG Dingguo 2 (1. Department of Mathematics, Shihezi University, Shihezi, Xinjiang, , P. R. China; 2. Department of Mathematics, Qufu Normal University, Qufu, Shandong, , P. R. China) Abstract: This paper gives some properties of T-comonads similar to A-corings(A is an algebra, T is a monad). First, it introduces some relative definitions of firm monad. Then it studies the equivalent propositions of Frobenius functors. Finally, it gives five equivalent propositions of separable T-comonads. Key words: T-comonads; separable functor; Frobenius functor

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